Intermittency in chaotic rotations

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 62, Issue 1 A, 2000, Pages

  Lai, Ying Cheng ^a,b   Armbruster, Dieter ^a   Kostelich, Eric J ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

^b Arizona State University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; PHASE SPACE METHODS; ROTATIONAL FLOW;

CHAOTIC ROTATIONS;

STATISTICAL MECHANICS;

EID: 0034225728     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.62.R29     Document Type: Article

References (33)

1. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
2. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
3. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
4. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
5. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
6. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
7. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
8. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
9. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996); 78, 4193 (1997); U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996); A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997); E. Rosa, E. Ott, and M. H. Hess, ibid. 80, 1642 (1998); K. J. Lee, Y. Kwak, and T. K. Lim, ibid. 81, 321 (1998); B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999); M. A. Zaks, E.-H. Park, M. G. Rosenblum, and J. Kurths, Phys. Rev. Lett. 82, 4228 (1999); V. Andrade, R. Davidchak, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).
10. A. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
11. T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 79, 3885 (1997).
12. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
13. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
14. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
15. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
16. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
17. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
18. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
19. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
20. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
21. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
22. See, for example, E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); 75, 1087 (1986); H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992); A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); F. Rödelsperger, A. Čenys, and H. Benner, Phys. Rev. Lett. 75, 2594 (1995); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
23. O. E. Rössler, Phys. Lett. A 71, 155 (1979).
24. S. C. Vankataramani, T. M. Antonsen, E. Ott, and J. C. Sommerer, Physica D 96, 66 (1996).
25. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
26. D. Gabor, J. IEE London 93, 429 (1946); P. Panter, Modulation, Noise, and Spectral Analysis (McGraw-Hill, New York, 1965).
27. D. Gabor, J. IEE London 93, 429 (1946); P. Panter, Modulation, Noise, and Spectral Analysis (McGraw-Hill, New York, 1965).
28. In general, there is no guarantee that a chaotic flow possesses a unique center of rotation, although some well studied chaotic systems such as the Rössler oscillator do. If there is no apparent unique center of rotation, a proper change of variable may lead to one, as illustrated by the example with the Lorenz oscillator. Generally, one can make use of the procedure described in Ref. [11] to decompose a chaotic flow into distinct modes, each with a unique center of rotation.
29. D. Marthaler, D. Armbruster, Y.-C. Lai, and E. J. Kostelich (unpublished).
30. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, Proc. R. Soc. London, Ser. A 454, 903 (1998).
31. L. O. Chua, M. Komuro, and T. Matsumoto, IEEE Trans. Circuits Syst. 33, 1073 (1986); R. Madan, Chua's Circuit: A Paradigm for Chaos (World Scientific, Singapore, 1993).
32. L. O. Chua, M. Komuro, and T. Matsumoto, IEEE Trans. Circuits Syst. 33, 1073 (1986); R. Madan, Chua's Circuit: A Paradigm for Chaos (World Scientific, Singapore, 1993).
33. A widely used technique in the study of chaotic flows is the Poincaré surface-of-section technique. On a Poincaré surface of section, the dynamics can be described by a discrete map whose phase-space dimension is one less than that of the original continuous flow. Chaotic flows can then be understood based on concepts that are convenient for maps such as unstable periodic orbits. The sectioning technique, however, suffers a fundamental drawback: the discrete map produced by it contains no information about the phase of the underlying flow. The framework laid out in this paper, which is based on the idea that chaos is organized around rotations, may thus provide additional insights about chaotic systems which are not revealed by Poincaré maps.